The rudiments of the non-relativistic quantum theory are here worked out, in preparation for its thorough application to atomic and molecular physics developed in the next chapters. While the formal architecture will rely on a number of statements, definitions, and mathematical properties introduced either for self-consistency and for further convenience, its physical foundation is found for comparison to the phenomenological and conceptual achievements discussed in the previous chapter. All properties here reported—although representing just a summary of quantum mechanics—are discussed together with their formal proof, so as to provide good training for getting familiar with the quantum formalism.
A new field of research centered on quantum science and technology is emerging. This rapidly developing field deals with the study and control of single quantum entities, i.e. atoms, ions, photons, which are no longer isolated but are parts of a larger system. Manipulation of these quantum entities forms the basis of modern quantum technology, from quantum computation to quantum communication and quantum sensing. One, therefore, needs to reconsider the basic concepts of state, evolution, and measurement for these so-called open quantum systems. This is a pedagogical review paper on the rudiments of modern quantum mechanics, which keeps citations to the literature to a minimum and assumes only a knowledge of elementary quantum mechanics. It is meant to prepare the reader for studying the modern research papers at the forefront of this field